Lemma 23.1.20. Let \(p\colon E \to C\) be a cocartesian fibration. Then the following conditions are equivalent:
- (1)
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The functor \(p\) is also a cartesian fibration.
- (2)
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For every morphism \(f\colon X \to Y\) in \(C\), the cocartesian transport functor \(f_!\colon E_X \to E_Y\) admits a right adjoint \(f^*\colon E_Y \to E_X\).
Proof. The cartesian lift of \(f\colon X\to Y\) ending at \(\widetilde {Y}\in E_Y\) is obtained by composing the cocartesian lift \(f^*\widetilde {Y}\to f_!f^*\widetilde {Y}\) with the counit \(f_!f^*\widetilde {Y}\to \widetilde {Y}\). The universal property, as well as the converse construction of the adjunction from cartesian lifts, is proved in Reference ? of [Cisinski et al. (2026)]. □
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